Statistical mechanics of warm and cold unfolding in proteins

Abstract

We present a statistical mechanics treatment of the stability of globular proteins which takes explicitly into account the coupling between the protein and water degrees of freedom. This allows us to describe both the cold and the warm unfolding, thus qualitatively reproducing the known thermodynamics of proteins. Classification: BiophysicsThe folded conformation of globular proteins is a state of matter peculiar in more than one respect. The density is that of a condensed phase (solid or liquid), and the relative position of the atoms is, on average, fixed; these are the characteristics of the solid state. However, solids are either crystalline or amorphous, and proteins are neither: the folded structure, while ordered in the sense that each molecule of a given species is folded in the same way, lacks the translational symmetry of a crystal. In Schrödinger's words, proteins are "aperiodic crystals". Unlike any other known solids, globular proteins are not really rigid, being able to perform large conformational motions while retaining locally the same folded structure. Finally, these are mesoscopic systems, consisting of a few thousand atoms.Quantitatively, the peculiarities of this state of matter are perhaps best appreciated from the thermodynamics. Delicate calorimetric measurements [1-3] on the folding transition of globular proteins reveals the following picture: first the transition is first order, at least in the case of single domain proteins. Secondly, the stability of the folded state, i.e. the difference in Gibbs potential ∆G between the unfolded and the folded state is at most a fraction of kT room per aminoacid. Following Privalov [3], we will refer to this property as "cooperativity". The Gibbs potential difference ∆G, as a function of temperature, is non monotonic: it has a maximum around room temperature (where ∆G > 0 and so the folded form is stable), then crosses zero and becomes negative both for higher and lower temperatures. Correspondingly, the protein unfolds not only at high, but also at low temperatures. This phenomenon of "cold unfolding", which is observed experimentally, is most peculiar: solids usually do not melt upon cooling! For temperatures around the cold unfolding transition and below, the enthalpy difference ∆H between the unfolded and the folded state is negative; this means that cold unfolding proceeds with a release of heat (a negative latent heat), as is also observed experimentally; at the higher unfolding transition, on the contrary, ∆H > 0 which corresponds to the usual situation of a positive latent heat. Fig.1 shows Privalovs measurements of the specific heat of myoglobin [3]. There are two peaks in the specific heat, corresponding to the two unfolding transitions, and a large gap ∆C in the specific heat between the unfolded and the folded state. This gap is again peculiar to proteins: usually, for a melting transition ∆C ≈ 0 (e.g. for ice at 0 C C = 1.01 cal/gK while for water at 0 C C = 1.00 cal/gK). The existence of this gap ∆C is related to the phenomenon of cold...

Thermodynamical Implications of a Protein Model with Water Interactions

“…This is in agreement with experimental data, and opposite to the situation found in the earlier protein models of Refs. [8,9,11]. …”

“…φ i = 0 means that node i is open (unfolded), and φ i = 1 means that node i is folded. Assuming N nodes, a Hamiltonian (H 1 ) for the energies associated to the polypeptide chain is in a compact way written [8,9,10,11] …”

“…This is termed as the "hydrophobic force", and the water-protein interaction is incorporated by an energy ladder representing each individual water molecule associated to the unfolded parts of the protein (i.e. all nodes where φ i = 0) [8,9,11] …”

Thermodynamics of Heat-Shock Response

“…Thus DG G͑folded͒ 2 G͑unfolded͒ connected to the folded state of a protein is at a minimum at T 0 . The corresponding maximum of stability is the result of a balance between destabilization from entropy of polymer degrees of freedom at high T and destabilization due to decreased entropic contribution to hydrophobicity at low T [12,13]. One should also expect similar behavior for some parts of a protein [13], and thus expect a maximum binding for hydrophobic protein-protein associations at about T 0 .…”

“…The corresponding maximum of stability is the result of a balance between destabilization from entropy of polymer degrees of freedom at high T and destabilization due to decreased entropic contribution to hydrophobicity at low T [12,13]. One should also expect similar behavior for some parts of a protein [13], and thus expect a maximum binding for hydrophobic protein-protein associations at about T 0 . Quantitatively, the size of the DG change inferred from the measured value of a 0.03K 22 corresponds to a changed G of about 20 30 kT (about 15 kcal͞mol), for a temperature shift of about 40 50 ± C. This matches the change observed for typical single domain proteins [12].…”